Theorem: Let $X$ be a manifold, $x \in X$ and let V be a vector field s.t. $V(x) \ne 0$. Then there exists a chart $(U,f)$ on $X$ such that $x \in U$ and for all $y \in U$, we have $\Delta_f (V(y)) = \frac{\partial}{\partial x_1}|_{f(y)}$
My question is: what on earth does $\frac{\partial}{\partial x_1}|_{f(y)}$ mean?
The way I see it, $\Delta_f$ is a map from $T_y X$ to $\mathbb{R}^n$ so $\frac{\partial}{\partial x_1}|_{f(y)}$ should be an element of $\mathbb{R}^n$. But I don't know how to interpret this object. My first thought was that we're taking the partial derivative of $f(y)$ with respect to $x_1$ but this would always be $0$ since $f(y)$ is a constant vector so this doesn't make sense.
Thank you.
The point of the statement is that you make the vector into $(1, 0, \ldots, 0),$ and this is what the formula really tells you.
The tangent space of $\mathbb{R}^n$ can be identified with $\mathbb{R}^n,$ but really it is spanned by vectors $\frac{\partial}{\partial x_i}$ (that is, it is a set of derivations), as for any manifold. $\frac{\partial}{\partial x_1}|_{f(y)}$ is one of the basis vectors - as a derivation, what it does is it takes the $x_1$-derivative and then plugs in the points $f(y)$.